Normality-small subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed normality-small if the only normal subgroup with which its product is the whole group, is in fact the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed normality-small if whenever $N\triangleleft G$ is such that $HN=G$, we have $N=G$.

In terms of the small operator

This property is obtained by applying the small operator to the property: normality
View other properties obtained by applying the small operator

The small operator takes as input a subgroup property and outputs the property of being a subgroup whose join with every proper subgroup having this property is proper. The subgroup property of being normality-small is obtained by applying the small operator to the subgroup property of normality.

Relation with other properties

Related properties

• Normality-large subgroup
• Strongly contranormal subgroup
• Frattini-embedded normal subgroup

Metaproperties

Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

Any subgroup sitting inside a normality-small subgroup is normality-small. This follows directly from the definition.